Prof. Rahul Mishra
M: 9999907099, 9818932244
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EDUCATION SOLUTION
MATHEMATICS OF CLASS IX
CHAPTER-2 POLYNOMIALS
Q.1. Which of the following expressions are polynomials in one variable and which are not? State
reason for your answer.
(i) 4x2 – 3x + 7
(iii) 3√𝒕 + t√𝟐
10
3
(ii) y2 + √𝟐
(iv) y +
𝟐
𝒚
50
(v) x + y + t
Q.2. Write the coefficient of x 2 in each of the following polynomials.
(i) 2 + x2 + x
(ii) 2 – x2 + x3
𝝅
(iii) x2 + x
𝟐
(iv) √𝟐 x – 1
Q.3. Give one example each of a binomial of degree 35 and monomial of degree 100.
Q.4. Write the degree of each of the following polynomials.
(i) 5x3 + 4x2 + 7x
(ii) 4 – y2
(iii) 5t - √𝟕
(iv) 3
Q.5. Classify the following as linear, quadratic and cubic polynomials.
(i) x2 + x
(ii) x – x3
(iii) y + y2 + 4
(iv) 1 + x
(v) 3t
(vi) r2
(vii) 7x3
Q.6. For the polynomial
𝟕𝒙𝟓 – 𝟓𝒙+𝟔 𝟑 4
- x – x11 . Write
𝟏𝟏
𝟒
(i)
The degree of the polynomial.
(ii)
The coefficient of x5 .
(iii)
The coefficient of x11 .
(iv)
The constant term.
1
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M: 9999907099, 9818932244
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Q.7. Find the coefficient of x 2 in (3x + x3 ) (𝒙 +
𝟏
𝒙
).
Q.8. If p = 10, then find the degree of the polynomial f(x) = (x – p)2 + 5.
Q.9. Verify that x = 1 is a zero of the polynomial 2x 2 – x – 1.
Q.10. Find a zero of the polynomial p(x) = 2x + 4.
Q.11. Verify whether 3 and 0 are zeroes of the polynomial x 2 – 3x.
Q.12. If the polynomials ax3 + 3x2 + 5x – 4 and x3 – 4x + a leave the same remainder, when divided
by (x-2) then find the value of a.
Q.13. Find the value of the polynomial 5x – 4x2 + 3 at
(i) x = 0
(ii) x = - 1
(iii) x = 2
Q.14. Find p(0), p(1) and p(2) for each of the following polynomials.
(i) p(y) = y2 – y + 1
(ii) p(t) = 2 + t + 2t2 – t3
(iii) p(x) = x3
Q.15. Verify whether the following are zeroes of the polynomial indicated against them.
(i) p(x) = 3x + 1, x = (ii) p(x) = 5x – 𝝅, x =
𝟏
𝟑
𝟒
𝟓
(iii) p(x) = x2 – 1, x = 1, -1
Q.16. Find the zero of the polynomial in each of the following cases.
(i) p(x) = x + 5
(ii) p(x) = x – 5
(ii) p(x) = 2x + 5
(iv) p(x) = 3x – 2
Q.17. Find the remainder when x 3 + 3x2 + 3x + 1 is divided by
𝟏
(i) x + 1
(ii) x -
(iii) x
(iv) x + 𝝅
𝟐
Q.18. Find the remainder when x 3 – ax2 + 6x – a is divided by (x – a).
Q.19. Check, whether (7 + 3x) is a factor of 3x 3 + 7x.
Q.20. If x = 3 and x = 0 are zeroes of the polynomial 2x 3 – 8x2 + ab + b, then find the values of a and
b.
Q.21. The polynomial bx3 + 3x2 – 3 and 2x3 – 5x + b, when divided by (x – 4) leaves the remainder r1
and r2, respectively. Find the value of b, if 2r1 – r2 = 0.
Q.22. The polynomial
2
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M: 9999907099, 9818932244
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P(x) = x4 – 2x3 + 3x2 – ax + 3a – 7,
When divided by (x + 1), leaves the remainder 19. Find the value of a. Also, find the remainder,
when p(x) is divided by x + 2.
Q.23. If the polynomials p(x) = 2x 3 + bx2 + 3x – 5 and q(x) = x3 + x2 – 4x – b leaves the same
remainder, when divided by x – 2, then prove that b =
−𝟏𝟑
𝟓
.
Q.34. If the polynomials (2x 3 + kx2 + 3x – 5) and (x3 + x2 – 2x + k) leave the same remainder when
divided by (x – 3), find the value of k. Also, find the remainder in first case.
Q.35. Using factor theorem, show that (x + 1) is a factor of x 19 + 1.
Q.36. Factorise 2x2 + 7x + 3 by splitting the middle terms.
Q.37. Factorise x2 – 5x + 6 by using factor theorem.
Q.38. Factorise 2x3 – 5x2 – 19x + 42.
Q.39. Write (2x + 3y – 5z)2 in expanded form.
Q.40. Evaluate (999)3 by using suitable identities.
Q.41. Evaluate 105 × 106 without multiplying directly.
Q.42. Determine which of the following polynomial has (x + 1) as a factor ?
(i) x3 + x2 + x + 1
(ii) x4 + x3 + x2 + x + 1
(iii) x4 + 3x3 + 3x2 + x + 1
Q.43. Find the value of k, if (x-1) is a factor of p(x) in each of the following cases.
(i) p(x) = x2 + x + k
(ii) p(x) = 2x2 + kx + √𝟐
(iii) p(x) = kx2 - √𝟐 x + 1
Q.44. Factorise
(i) 12x2 - 7x + 1
(ii) 2x2 + 7x + 3
(iii) 6x2 + 5x – 6
(iv) 3x2 – x – 4
Q.45. Factorise the following polynomial by remainder theorem.
(i) x3 – 2x2 – x + 2
(ii) x3 – 3x2 – 9x – 5
(iii) x3 + 13x2 + 32x + 20
Q.46. Use suitable identities to find the following products.
(i) (x + 4)(x + 10)
(ii) (x + 8)(x – 10)
3
Prof. Rahul Mishra
M: 9999907099, 9818932244
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𝟑
𝟑
𝟐
𝟐
(iv) (𝒚𝟐 + ) (𝒚𝟐 − )
(iii) (3x + 4)(3x – 5)
Q.47. Evaluate the following products without multiplying directly.
(i) 103 × 107
(ii) 95 × 96
(iii) 104 × 96
Q.48. Factorise the following using appropriate identities.
(i) 9x2 + 6xy + y2
(iii) x2 -
(ii) 4y2 – 4y + 1
𝒚𝟐
𝟏𝟎𝟎
Q.49. Expand each of the following using suitable identities.
(i) (x + 2y + 4z) 2
(ii) (2x – y + z)2
(iii) (-2x + 3y + 2z)2
(iv) (3a – 7b – c)2
Q.50. Factorise
(i) 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz
(ii) 2x2 + y2 + 8z2 - 2√𝟐 xy + 4√𝟐 yz – 8xz
Q.51. Write the following cubes in expanded form.
(i) (2x + 1) 3
(ii) (2a – 3b)3
𝟑
𝟐
(iii) ( 𝒙 + 𝟏)3
(iv) (𝒙 − 𝒚)3
𝟐
𝟑
Q.52. Evaluate the following using suitable identities.
(i) (99) 3
(ii) (102) 3
(iii) (998) 3
Q.53. Factorise each of the following.
(i) 8a3 + b3 + 12a2b + 6ab2
(ii) 8a3 – b3 – 12a2b + 6ab2
(iii) 27 – 125a3 – 135a + 225a2
Q.54. Verify that
(i) x3 + y3 = (x + y)(x2 – xy + y2)
(ii) x3 – y3 = (x – y)(x2 + xy + y2 )
Q.55. Factorise each of the following.
(i) 27y3 + 125z3
(ii) 64m3 – 343n3
Q.56. Factorise 27x3 + y3 + z3 – 9xyz
Q.57. Verify that x3 + y3 + z3 – 3xyz
Q.58. If (x + y + z) = 0, then show that x3 + y3 + z3 = 3xyz.
Q.59. Without actually calculating the cubes, find the value of each of the following:
4
Prof. Rahul Mishra
M: 9999907099, 9818932244
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(i) (-12)3 + (7)3 + (5)3
(ii) (28) 3 + (-15)3 + (-13)3
Q.60. Give possible expressions for the length and the breadth of each of the following rectangles,
in which their areas are given.
(i) Area 25a2 – 35a + 12
(ii) Area 35y2 + 13y – 12
Q.61. What are the possible expressions for the dimensions of the cuboids whose volumes are
given below ?
(i) Volume 3x2 – 12x
(ii) Volume 12 ky2 + 8ky – 20k
Q.62. If (x – a) is a factor of 4x2 – mx – na, then prove that a =
𝒎+𝒏
𝟒
.
Q.63. Factorise 7x2 + 2√𝟏𝟒x + 2.
Q.64. If a + b + c = 9 and ab + bc + ca = 40, then find the value of a2 + b2 + c2 .
Q.65. Find the value of 27x3 + 8y3 , if 3x + 2y = 20 and xy =
𝟏𝟏
𝟗
.
Q.66. What is the degree of the polynomial √𝟓 ?
Q.67. Find the coefficient of x 2 in the expansion of (x – 4)2 .
Q.68. What is the degree of zero polynomial ?
Q.69. What type of the polynomial -2 is ?
Q.70. How many zeroes has a cubic polynomial ?
Q.71. What is the best way to evaluate (998) 2 ?
Q.72. If x + y + z = 0, then find the value of x 3 + y3 + z3 .
Q.73. What is the zero of the polynomial
p(x) = a2x, a ≠ 0 ?
Q.74. A zero polynomial has how many zeroes ?
Q.75. What is the zeroes of the polynomial p(x) = x2 + x – 6 ?
Q.76. Find the remainder, when p(x) = x 3 – ax2 + x is divided by (x - a).
Q.77. Find the factors of 3x 2 – x – 4.
Q.78. Which identity do we use to factorise x 2 -
𝒚𝟐
𝟏𝟎𝟎
?
Q.79. If x11 + 101 is divided by x + 1, then find the remainder.
𝒙
𝒚
𝒚
𝒙
Q.80. If + = - 1, x ≠ 0, y ≠ 0, then find the value of x 3 – y3 .
5
Prof. Rahul Mishra
M: 9999907099, 9818932244
www.vaishalieducationpoint.com, www.educationsolution.co
Q.81. If f(x) = x – 9, then find the value of f(x) – f(-x).
Q.82. Find the value of (1015) 2 – (1014)2.
Q.83. If (x + 2) and (x – 1) are factors of (x3 + 10x2 + mx + n), then find the values of m and n.
Q.84. What is the value of the expression 4a2 + b2 + 4ab + 8a + 4b + 4 ?
Q.85. What is the must condition for the indices of x in a polynomial of x ?
Q.86. If (x100 + 2x99 + k) is divisible by (x + 1), then find the value of k.
Q.87. Find the remainder, when p(x) = 4x 3 – 12x2 + 11x – 5 is divided by (2x -1).
𝟐
Q.88. If p(y) = 2y3 – y2 – 13y – 6, then find the value of p (− ).
𝟑
2
Q.89. If x + kx + 12 = (x – 6) (x – 2) for all x, then find the value of k.
Q.90. Find the factor of the expression ab + bc + ax + cx.
Q.91. What is the value of 305 × 308 ?
Q.92. The volume of a cuboid is 2x 2 – 16, then find its possible dimensions.
Q.93. Find the value of 583 – 243 – 343 .
Q.94. What is the degree of the polynomial
4x4 + 0x3 + 0x5 + 5x + 7 ?
Q.95. Find the remainder, when x 51 + 51 is divided by x + 1.
Q.96. If p(x) = x2 - 2√𝟐 x + 1, then find the value of p(2√𝟐).
Q.97. What is the factor of (x + y) 3 – (x3 + y3) ?
Q.98. Find the factors of (25x2 – 1) + (1 + 5x)2.
Q.99. If f(t) = 4t2 – 3t + 6, then find
(i) f(4)
(ii) f(-5)
Q.100. Verify that, 0 and 3 are the zeroes of the polynomial p(x) = x 2 – 3x.
Q.101. Find the remainder when the polynomial f(x) = 4x 3 – 12x2 + 14x – 3 is divided by 2x – 1.
Q.102. Find the remainder, when
f(x) = x3 – 6x2 + 13x + 60 is divided by (x + 2).
Q.103. Find the remainder, when
f(x) = x3 - ax2 + 2x – a is divided by (x – a).
Q.104. Find the value of a for which (x – a) is a factor of the polynomial f(x) = x 5 – a2x3 + 2x + a – 3.
Q.105. Show that (x + 5) is a factor of the polynomial f(x) = x 3 + x2 + 3x + 115.
Q.106. Factorise
6
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M: 9999907099, 9818932244
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(i) 5a (b + c) – 7b(b + c)
(ii) 6(2a + 3b) 2 – 8(2a + 3b).
Q.107. Simplify √𝟐𝒂𝟐 + 𝟐 √𝟔 𝒂𝒃 + 𝟑𝒃𝟐.
𝟏
𝟏
𝟐
𝟒
Q.108. Expand ( 𝒂 − 𝒃 + 𝟐)2 .
Q.109. Factorise the following expressions
(i) 25x2 + 4y2 + 9z2 – 20xy – 12yz + 30xz
(ii) 9x2 + 16y2 + 4z2 – 24xy + 16yz – 12xz
Q.110. If 𝒂𝟐 +
𝟗
𝟑
𝒂𝟐
= 31, then what is the value of a - ?
𝒂
Q.111. By actual division, find the quotient and the remainder when the first polynomial x 4 + 1 is
divided by the second polynomial x – 1.
𝒙
𝒚
𝒚
𝒙
Q.112. What must be subtracted from to make it ?
Q.113. If x2 +
𝟏
𝒙𝟐
= 14, then find x3 +
𝟏
𝒙𝟑
.
Q.114. Simplify the following expressions
(i) (x + y + z) 2 + (x + y – z)2
(ii) (2x + p – c)2 – (2x – p + c)2
Q.115. Using remainder theorem, find the remainder when q(x) = x 4 – 2x2 + 6x + 3 is divided by
(x – 2).
Q.116. If the polynomials (2x 3 + ax2 + 3x – 5) and (x3 + x2 – 2x + a) leave the same remainder when
divided by (x – 2), find the value of a. Also, find the remainder in each case.
Q.117. The remainder of the polynomial 5 + bx – 2x2 + ax3 , when divided by x – 2 is twice the
remainder when it is divided by (x + 1). Show that 10a + 4b = 9.
Q.118. For what value of k is the polynomial (2x 4 + 3x2 + 2kx2 + 3x + 6) exactly divisible by (x + 2) ?
Q.119. Expand
(i) (0.1x – 0.2y)3
𝒚
𝒙
𝟑
𝟐
(ii) ( − )2
Q.120. If √𝒖 + √𝒗 - √𝒘 = 0, then find the value of (u + v – w).
𝟏
Q.121. If f(x) = x2 – 5x + 1, then evaluate f(2) – f(-1) + f ( ).
𝟑
Q.122. Find y2 +
𝟏
𝒚𝟐
and 𝒚𝟒 +
𝟏
𝒚𝟒
𝟏
, if y - = 9.
𝒚
Q.123. Factorise (p – q)3 + (q – r)3 + (r – p)3
Q.124. Find the product
7
Prof. Rahul Mishra
M: 9999907099, 9818932244
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(3x – 5y – 4) (9x2 + 25y2 + 15xy + 12x – 20y + 16).
Q.125. Factorise
a3 (b – c)3 + b3 (c – a)3 + c3 (a – b)3.
Q.126. Factorise
(5a – 7b)3 + (9c – 5a)3 + (7b – 9c)3.
Q.127. If x + y + 4 = 0, then find the value of x 3 + y3 – 12xy + 64.
Q.128. Find the value of k, if (x – 1) is a factor of p(x) = kx2 - √𝟐 x + 1.
Q.129. Evaluate
(i) (104)3
(ii) (999) 3
Q.130. If x – y = 5 and xy = 84, then find the value of x 3 – y3 .
Q.131. Give an example of a polynomial which is
(i) monomial of degree 1.
(ii) binomial of degree 22.
(iii) trinomial of degree 5.
Q.132. Without actual division, prove that (2x 4 – 6x3 + 3x2 + 3x – 2) is exactly division by (x2 – 3x +
2).
Q.133. If (x3 + ax2 + bx + 6) has (x – 2) as a factor and leaves a remainder 3 when divided by (x – 3),
then find the values of a and b.
Q.134. If x = 2y + 6, then find the value of x 3 – 8y3 – 36xy – 216.
Q.135. Prove that x3 + y3 + z3 – 3xyz
Q.136. If a,b and c are all non-zero and a + b + c = 0, then prove that
𝒂𝟐
𝒃𝒄
+
𝒃𝟐
𝒄𝒂
+
𝒄𝟐
𝒂𝒃
= 3.
Q.137. Factorise x3 + 13x2+ 32x + 20.
Q.138. If (x + a) is a factor of the polynomials x 2 + px + q and x2 + mx + n, then prove that a =
Q.139. Find the value of
𝟏
𝟐𝟕
𝒏−𝒒
𝒎−𝒑
.
𝒓
r3 – s3 + 125t3 + 5rst, when s = + 5t.
𝟑
Q.140. If x + y + z = 1, xy + yz + zx = -1 and xyz = -1, then find the value of x 3 + y3 + z3 .
Q.141. Let A and B be the remainder, when the polynomials y 3 + ay2 – 12y + 6 are divided by (y + 1)
and (y – 2), respectively. If 2A + B = 6, then find the value of a.
Q.142. Factorise a7 – ab6.
Q.143. Factorise (x + y)3 – (x – y)3 .
Q.144. Find the square root of
8
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M: 9999907099, 9818932244
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(x2 – 5x + 6)2 – (x2 – 6x + 8)2.
Q.145. Find the zeroes of the polynomial
p(x) = (x – 2 )2 – (x + 2)2.
Q.146. Factorise 2y4 + y3 – 14y2 – 19y – 6.
Q.147. If a teacher Divides a material of volume (x3 + 6x3 + 12x + 8) cubic units among three
students of his class equally. Is it possible, to find the quantity of material each get and which
moral value is depicted ?
Q.148. Two friends start business together. They decided to share their capitals depending upon a
variable expenditure. The capital polynomial of the two partners together is given by polynomial
6x2 + 11x – 35. Which is the product of their individual share factors. (i) Find their factors. (ii) Are
their capital shares same ? (iii) Write the value depicted by this question.
Q.149. Factorise 9x2 + y2 + z2 – 6xy + 2yz – 6xz. Hence, find its value when x = 1, y = 2 and z = -1.
Q.150. Factorise a12 y4 – a4y12 .
Q.151. Write the coefficient of x in the expansion of (x + 5) 3 .
Q.152. Find the zeroes of the polynomial p(x) = (x – 2)2 – (x + 2)2 .
𝟏
Q.153. Find the value of p( ) for p(x) = x4 – x2 + x.
𝟐
Q.154. Find the value of the polynomial at the indicated value of variable p(x) = 3x 2 – 4x + √𝟏𝟏 at x
= 2.
Q.155. Find the zeroes of x2 – 7.
Q.156. Verify that 1 is not a zero of the polynomial 4y 4- 3y3 + 2y2 – 5y + 1.
Q.157. Find the zero of the polynomial p(x), where p(x) = ax + 3, a ≠ 0.
Q.158. Find the value of p for which x + p is a factor of x 2 + px + 3 – p.
Q.159. Find the value of 94 × 96, by using identity.
𝟏
𝟏
𝒑
𝒑𝟐
Q.160. If p + = 3, then find the value of p2 +
.
Q.161. What are the possible expression for the dimensions of a cuboid, whose volume is given
below ?
Volume = 12ky2 + 8ky – 20k
Q.162. Find the value of x3 – 8y3 – 36xy – 216, when x = 2y + 6.
Q.163. Find the value of 64x 3 + 125z3, if 4x + 5z = 19 and xz = 5.
Q.164. If a + b = 10 and a2 + b2 = 58, then find the value of a3 + b3 .
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